PHYSICAL REVIEW B 88, 144420 (2013)

Longitudinal magnetization reversal in ferromagnets with Heisenberg exchange and strongsingle-ion anisotropy

E. G. Galkina,1,2 V. I. Butrim,3 Yu. A. Fridman,3 B. A. Ivanov,2,4,5,* and Franco Nori2,6

1Institute of Physics, 03028 Kiev, Ukraine2CEMS, RIKEN, Saitama, 351-0198, Japan

3Vernadsky Taurida National University, 95007 Simferopol, Ukraine4Institute of Magnetism, 03142 Kiev, Ukraine

5Taras Shevchenko National University of Kiev, 01601 Kiev, Ukraine6Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

(Received 30 June 2013; revised manuscript received 22 September 2013; published 23 October 2013)

We analyze theoretically the novel pathway of ultrafast spin dynamics for ferromagnets with high enoughsingle-ion anisotropy. This longitudinal spin dynamics includes the coupled oscillations of the modulus of themagnetization together with the quadrupolar spin variables, which are expressed through quantum expectationvalues of operators bilinear on the spin components. Even for a simple single-element ferromagnet, such dynamicscan lead to a magnetization reversal under the action of an ultrashort laser pulse.

DOI: 10.1103/PhysRevB.88.144420 PACS number(s): 75.10.Jm, 75.10.Hk, 78.47.J−, 05.45.−a

I. INTRODUCTION

Which is the fastest way to reverse the magnetization ofeither a magnetic particle or a small region of a magneticfilm? This question has attracted significant interest, bothfundamental and practical, for magnetic information storage.1

Intense laser pulses, with durations less than a hundredfemtoseconds, are able to excite the ultrafast evolution of thetotal spin of a magnetically ordered system on a picosecondtime scale, see, e.g., the reviews.2–4 The limitations for the timeof magnetization reversal come from the characteristic featuresof the spin evolution for a magnet with a concrete type ofmagnetic order. The dynamical time cannot be shorter than thecharacteristic period of spin oscillations T , T = 2π/ω0, whereω0 is the magnetic resonance frequency. For ferromagnets,the frequency of standard spin oscillations (precession) isω0,FM = γHr, where γ is the gyromagnetic ratio, and Hr isan effective field of relativistic origin, like the anisotropy field,which is usually less than a few Tesla. Thus, the dynamical timefor Heisenberg ferromagnets cannot be much shorter than 1ns. For antiferromagnets, all the dynamical characteristics areexchange enhanced, and ω0,AFM = γ

√HexHr, where Hex is

the exchange field, Hex = J/2μB, J is the exchange integral,and μB is the Bohr magneton, see Ref. 5. The excitation of ter-ahertz spin oscillations has been experimentally demonstratedfor transparent antiferromagnets using the inverse Faradayeffect or the inverse Cotton-Mouton effect.6–11 The nonlinearregimes of such dynamics include the inertia-driven dynamicalreorientation of spins on a picosecond time scale, which wasobserved in orthoferrites.10,11

The exchange interaction is the strongest force in mag-netism, and the exchange field Hex can be as strong as 103

T. The modulus of the magnetization is determined by theexchange interaction, and the direction of the magnetization isgoverned by relativistic interactions. It would be very temptingto produce a magnetization reversal by changing the modulusof the magnetization vector, i.e., via the longitudinal dynamicsof M. For such a process, dictated by the exchange interaction,the characteristic times could be of the order of the exchange

time τex = 1/γHex, which is shorter than 1 ps. Nevertheless,within the standard approach such dynamics is impossible. Theevolution of the modulus of the magnetization M = |M| withinthe closed Landau-Lifshitz equation for the magnetization only(or the set of such equations for the sublattice magnetizationsMα) is purely dissipative.12 This feature could be explainedas follows: Two angular variables θ and ϕ describing thedirection of the vector M within the Landau-Lifshitz equationdetermine the pair of conjugated Hamilton variables (cos θ

and ϕ are the momentum and coordinate, respectively). Also,the evolution of the single remaining variable M = |M|,governed by a first-order equation can be only dissipative;see a more detailed discussion below. Moreover, the exchangeinteraction conserves the total spin of the system, and therelaxation of the total magnetic moment of any magnet can bepresent only when accounting for relativistic effects. Thus therelaxation time for the total magnetic moment is relativisticbut it is exchange enhanced, as was demonstrated within theirreversible thermodynamics of the magnon gas.12 Note herethat the relaxation of the magnetization of a single sublatticefor multisublattice magnets can be of purely exchange origin.13

Recently, magnetization reversal on a picosecond time scalehas been experimentally demonstrated for the ferrimagneticalloy GdFeCo, see Refs. 13 and 14. These results can be ex-plained within the concept of exchange relaxation, developedby Baryakhtar,15 accounting for the purely exchange evolutionof the sublattice magnetization.16 Such an exchange relaxationcan be quite fast, but its characteristic time is again longer thanthe expected “exchange time” τex = 1/γHex.

Thus, the ultrafast mechanisms of magnetization reversalimplemented so far are: the dynamical (inertial) switching pos-sible for antiferromagnets,10,11 and the exchange longitudinalevolution for ferrimagnets.13,14,16 These are both quite fast,with a characteristic time of the order of picoseconds; but theircharacteristic times are longer than the “ideal estimate”: theexchange time τex.

In this work we present a theoretical study of the possibilityof the dynamical evolution of the modulus of the magnetizationfor ferromagnets with high enough single-ion anisotropy that

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GALKINA, BUTRIM, FRIDMAN, IVANOV, AND NORI PHYSICAL REVIEW B 88, 144420 (2013)

can be called longitudinal spin dynamics. For such a spinevolution, a dynamical magnetization reversal is possibleeven for a simple single-element ferromagnet. Longitudinaldynamics does not exist in Heisenberg magnets, and thisdynamics cannot be described in terms of the Landau-Lifshitzequation, or using the Heisenberg Hamiltonian, which isbilinear over the components of spin operators for differentspins, see more details below in Sec. II. The key ingredient ofour theory is the inclusion of higher-order spin quadrupolevariables. It is known that for magnets with atomic spinS > 1/2, allowing the presence of single-ion anisotropy,the spin dynamics is not described by a closed equationfor spin dipolar variable 〈S〉 alone (or magnetization M =−2μB〈S〉).17–24 Here and below 〈· · · 〉 means quantum and(at finite temperature) thermal averaging. To be specific, wechoose the spin-one ferromagnet with single-ion anisotropy,the simplest system allowing this effect. The full descriptionof these magnets requires taking into account the dynamics ofquadrupolarvariables Sik = (1/2)〈SiSk + SkSi〉 that representthe quantum averages of the operators, bilinear in the spin com-ponents. Such magnets are of significant interest, both fromthe viewpoint of fundamental science and for applications, ascan be seen from the recent reviews.25,26 Our theory is basedon the consistent semiclassical description of a full set of spinquantum expectation values (dipolar and quadrupolar) for thespin-one system, which was investigated by many authors fromdifferent viewpoints.17–24 As we will show, the longitudinaldynamics of spin, including nonlinear regimes, can be excitedby a femtosecond laser pulse. With natural accounting forthe dissipation, the longitudinal spin dynamics can lead tochanging the sign of the total spin of the system (longitudinalmagnetization reversal).

II. MODEL DESCRIPTION

The Landau-Lifshitz equation was proposed many yearsago as a phenomenological equation, and it is widely usedfor the description of various properties of ferromagnets.Concerning its quantum and microscopic basis, it is worthnoting that this equation naturally arises using the so-calledspin coherent states.27,28 These states can be introduced forany spin S as the state with the maximum value of thespin projection on an arbitrary axis n. Such states can beparametrized by a unit vector n; the direction of the lattercoincides with the quantum mean values for the spin operator〈S〉 = Sn (dipolar variables). This property is quite convenientfor linking the quantum physics of spins to a phenomenologicalLandau-Lifshitz equation. The use of spin coherent states ismost efficient when the Hamiltonian of the system is linearwith respect to the operators of the spin components. If aninitial state is described by a certain spin coherent state, itsquantum evolution will reduce to a variation of the parametersof the state (namely, the direction of the unit vector n), whichare described by the classical Landau-Lifshitz equation. Thus,spin coherent states are a convenient tool for the analysisof spin Hamiltonians containing only operators linear onthe spin components or their products on different sites.An important example is the bilinear Heisenberg exchangeinteraction, described by the first term in Eq. (1) below.

In contrast to the cases above, for the full description ofspin-S states, one needs to introduce SU(2S + 1) generalizedcoherent states.21–24 The analysis shows that spin coherentstates are less natural for the description of magnets whoseHamiltonian contains products of the spin component oper-ators at a single site. Such terms are present for magnetswith single-ion anisotropy or a biquadratic exchange inter-action. For such magnets, some nontrivial features, absent forHeisenberg magnets, are known. Among them we note thepossibility of so-called quantum spin reduction; namely thepossibility to have the value of |〈S〉| less than its nominalvalue |〈S〉| < S, even for pure states at zero temperature.This was mentioned by Moriya,29 as early as 1960. As anextreme realization of the effect of quantum spin reduction,we note the existence of the so-called spin nematic phaseswith a zero mean value of the spin in the ground state atzero temperature. In the last two decades, the interest onsuch states has been considerable, motivated by studies ofmulticomponent Bose-Einstein condensates of atoms withnonzero spin.30–33

A significant manifestation of quantum spin reduction isthe appearance of an additional branch of the spin oscillations,which is characterized by the dynamics (oscillations) of thelength of the mean value of spin without spin precession.17,19–24

The characteristic frequency of this mode can be quite high(of the order of the exchange integral). For this reason,for a description of resonance properties or thermodynamicbehavior of magnets, this mode is usually neglected, andthe common impression is that the dynamics of magneticmaterials with constant single-ion anisotropy K < (0.2–0.3)Jis fully described by the standard phenomenological theory.However, for an ultrafast evolution of the spin system under afemtosecond laser pulse, one can expect a lively demonstrationof this longitudinal high-frequency mode. Thus, it is importantto explore the possible manifestations of the effects of quantumspin reduction in the dynamic properties of ferromagnets.

The simplest model allowing spin dynamics with effects ofquantum spin reduction is described by the Hamiltonian

H = −1

2

∑n,�

JSnSn+� + K

2

∑n

(Sn,x)2, (1)

where Sn is the spin-one operator at the site n; J > 0 is theexchange constant for nearest-neighbors �, and K > 0 isthe constant of the easy-plane anisotropy with the plane yz

as the easy plane. The quantization axis can be chosen parallelto the z axis and 〈S〉 = 〈Sz〉ez. For the full description of spinS = 1 states, let us introduce SU(3) coherent states21–24

|u,v〉 =∑

j=x,y,z

(uj + ivj )|ψj 〉, (2)

where the states |ψj 〉 determine the Cartesian states forS = 1 and are expressed in terms of the ordinary states{|±1〉, |0〉} with given projections ±1, 0 of the operatorSz by means of the relations |ψx〉 = (|−1〉 − |+1〉)/√2,|ψy〉 = i(|−1〉 + |+1〉)/√2, |ψz〉 = |0〉, with the real vectorsu and v subject to the constraints u2 + v2 = 1, u · v = 0.All irreducible spin averages, which include the dipolarvariable 〈S〉 (average value of the spin) and quadrupoleaverages Sik , bilinear over the spin components, can be written

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through u and v as follows:

〈S〉 = 2(u × v),

Sik = 1

2〈SiSk + SkSi〉 = δik − uiuk − vivk. (3)

At zero temperature and within the mean-field approximation,the spin dynamics is described by the Lagrangian24

L = −2h∑

n

vn(∂un/∂t) − W (u,v), (4)

where W (u,v) = 〈u,v|H |u,v〉 is the energy of the system.We are interested in spin oscillations which are uniform in

space, and hence we assume that the discrete variables u andv have the same values for all spins and are only dependenton time. The frequency spectrum of linear excitations, whichconsists of two branches, can be easily obtained on the basis ofthe linearized version of the Lagrangian (4). In the general case,the system of independent equations for u and v, taking intoaccount the aforementioned constrains u2 + v2 = 1, u · v = 0,consists of four nonlinear equations, describing two differentregimes of spin dynamics. One regime is similar to thatfor an ordinary spin dynamics treated on the basis of theLandau-Lifshitz equation; it corresponds to oscillations of thespin direction. The second regime corresponds to oscillationsof the modulus of the magnetization 〈S〉 = S(t)ez, with thevectors u and v rotating in the xy plane perpendicularto 〈S〉. This mode of the spin oscillations corresponds tothe longitudinal spin dynamics. It is convenient to considerthese two types of dynamics separately. Particular nonlinearlongitudinal solutions, with 〈S〉 = s(t)ez and uz = 0, vz = 0,were found in Refs. 34 and 35. Note here that the longitudinaldynamics is much faster than the standard transversal one, andthe standard spin precession (described by the Landau-Lifshitzequation) at a picosecond time scale just cannot develop.Therefore, these two regimes, longitudinal and transverse, canbe treated independently, and we limit ourselves only to thelongitudinal dynamics with 〈S〉 = s(t)ez and uz = 0, vz = 0.

III. LONGITUDINAL SPIN DYNAMICS

To describe the longitudinal spin dynamics, it is convenientto introduce new variables: the spin modulus s = 2|u||v| =2uv and angular variable γ , with

u = u(ex cos γ − ey sin γ ), v = v(ex sin γ + ey cos γ ).

(5)

In this representation 〈Sz〉 = s, and the nontrivial quadrupolarvariables are

〈SxSy + SySx〉 =√

1 − s2 sin 2γ,⟨S2

y − S2x

⟩ =√

1 − s2 cos 2γ, (6)

with all other quantum averages being either zero (as thetransverse spin components 〈Sx,y〉 or Sxz, Syz) or trivial,independent on s and γ , as 〈S2

z 〉 = 1. The mean-field energy,written per one spin through the variables s, γ , takes the form

W (s,γ ) = −J

2s2 − K

4

√1 − s2 cos 2γ, (7)

FIG. 1. (Color online) The ground state parameters 〈S2x 〉 and 〈S2

y 〉(blue dash-dot line) and s = 〈Sz〉 (red solid line) as the function ofκ . In the lower part of the figure the graphic presentation of thevariables 〈S2

x 〉 and 〈S2y 〉 showing the shape of the cross section of the

quadrupolar ellipsoid in the xy plane for four values of κ; from leftto right κ = 0, 0.25, 0.5, and 0.75.

where J = JZ, Z is the number of nearest neighbors. Theground state at

√1 − s2 > 0 corresponds to cos 2γ = 1, with

the mean value of the spin s = ±s, s = √1 − κ2 < 1, that

is a manifestation of quantum spin reduction at nonzeroanisotropy. Here we introduce the dimensionless parameterκ = K/4J .

The values of the variables in the ground state are thefollowing:

〈Sz2〉 = 1, 〈Sx

2〉 = 1

2(1 + κ), 〈Sy

2〉 = 1

2(1 − κ). (8)

The dependencies on κ for the ground state parameters 〈S2x 〉,

〈S2y 〉, and 〈Sz〉, together with the schematic graphic image of

the xy cross section of the quadrupolar ellipsoid are shown inFig. 1. The quadrupolar variables can be graphically illustratedby using a quadrupolar ellipsoid, which is a three-axialellipsoid with the directions of the main axis (chosen to have〈S1S2〉 = 0): e3 = ez and e1, e2 and with the half-axes of theellipsoid equal to 〈S2

1 〉, 〈S22 〉, and 〈S2

3〉 = 〈S2z 〉 = 1. This is a

typical picture of ferro-quadrupolar ordering, known for manymagnets,25 and we are dealing with the coupled dynamics ofspin dipolar variables and quadrupolar variables. For κ � 1,the state with zero magnetization (quadrupolar state), and withthe values 〈Sz

2〉 = 〈Sx2〉 = 1 and 〈Sy

2〉 = 0, is realized.For the variables s and γ , the Lagrangian can be written as

L = hs∂γ

∂t− W (s,γ ), (9)

and hs and γ play the role of canonical momentum andcoordinate, respectively, with the Hamilton function W (s,γ ).The physical meaning of the above formal definitions is quiteclear: The angular variable γ describes the transformation ofthe quadrupolar variables under rotation around the z axis,

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with hs being the projection of the angular momentum on thisaxis. The Hamiltonian equations are

h∂γ

∂t= ∂W (s,γ )

∂s, h

∂s

∂t= −∂W (s,γ )

∂γ. (10)

The possible stationary solutions of the dynamical equations(10) (s = const, γ = const) are determined by the station-ary points of the energy function, i.e., by the conditions∂W (s,γ )/∂s = 0 and ∂W (s,γ )/∂γ = 0. The minimum of theenergy corresponds to the ground state of the system. Note thepresence of other stationary solutions, at s = 1, cos 2γ = 0;and at s = 0, sin 2γ = 0, which are unstable. These stationarypoints of the energy manifest themselves in the phase plane assaddle points and as singular phase trajectories at s = 1.

A. Small oscillations

Let us now start with the description of the dynamics ofsmall-amplitude oscillations. After the linearization aroundthe ground state, the equation leads to a simple formula for thefrequency of longitudinal oscillations

hωl = 2J s = 2J√

1 − κ2, (11)

which are in fact coupled oscillations of the projection of thespin and quadrupolar variables, see Fig. 2.

FIG. 2. (Color online) Graphic presentation of the variables s andγ and their evolution. The thick red arrow represents the mean valueof the spin. The quadrupolar variables are shown by the quadrupolarellipsoid, see the text, drawn in blue. (a) The ground state and (b)the standard transverse dynamics, i.e., the spin precession. (c)–(e)Transient values of the variables for longitudinal oscillations. (c) and(e) Correspond to the longest and shortest length of the spin, and atthe moment depicted in (d) the spin length equals to its equilibriumvalue, but the quadrupolar ellipsoid is turned on the angle γ withrespect to the x axis. (c)–(e) The shape of the unperturbed ellipsoidis shown in light gray.

One can see that, for a wide range of values of the anisotropyconstant, like κ < 0.2–0.8, this frequency ωl is of the order of(1.8–1.2)J/h, i.e., ωl is comparable to the exchange frequencyJ/h. Thus the longitudinal spin dynamics is expected to bequite fast. In contrast, standard transversal oscillations for apurely easy-plane model (1) are gapless (they acquire a finitegap when accounting for a magnetic anisotropy in the easyplane, which is usually small). Thus the essential difference inthe frequencies of these two dynamical regimes is clearly seen.

At a first glance, there is a contradiction between the con-cept of longitudinal dynamics caused by single-ion anisotropyand the result present in Eq. (11): The value of ωl is still finitefor vanishing anisotropy constant K; and it is even growing tothe value 2J/h when κ → 0. This can be explained as follows:In the limit κ → 0 the spin-quadrupole moment is simply zeroaccording to the formula (6). Hence the free rotation of thequadrupole ellipsoid is a free rotation of a quantity of zerosize, and the actual resonance does not exist in this limit.More formally, on a phase plane with coordinates (s,γ ) thisdynamics is depicted by vertical straight lines parallel to theγ axis, and the mean value of spin does not change, as can beseen by comparing Figs. 3(a) and 3(c) below. We will discussthis feature in more detail later with the analysis of nonlinearoscillations.

B. Nonlinear dynamics and phase plane analysis

Before considering damped oscillations, it is instructive todiscuss dissipationless nonlinear longitudinal oscillations. It isconvenient to present an image of the dynamics as a “phaseportrait” on the plane momentum-coordinate (s, γ ), whichshows the behavior of the system for arbitrary initial condi-tions. The phase trajectories in the plane without dissipationcan be found from the condition W (s,γ ) = const.

The energy (7) has an infinite set of minima, with s = ±s

and γ = πn, with equal energies (green ellipses in Fig. 3),and an infinite set of maxima at s = 0 and γ = π/2 + πn (redellipses in Fig. 3), here n is an integer. Only the minima withs = s and s = −s are physically different; equivalent extremeswith different values of n correspond to the equal values of theobservables and are completely equivalent. The minima on thephase plane correspond to foci with two physically differentequilibrium states with antiparallel orientation of spin s = ±s,and 〈S2

y − S2x〉 = √

1 − s2, 〈S2z 〉 = 1, 〈SxSy + SySx〉 = 0. The

saddle points are located at the values s = 0 and γ = πn.The lines with s = ±1 are singular; these correspond todegenerate motion with γ linear in time γ = ±tJ/h; the pointsat these lines where dγ /dt change sign can be treated asnonstandard saddle points.

The shape of the phase trajectories, i.e., the characteristicfeatures of oscillations, varies when changing the anisotropyparameter κ . Note first the general trend, the relative amplitudeof the changes of the spin and γ depends on κ: The biggerκ is, the larger values of the change of spin are observed.The topology of the phase trajectories change at the criticalvalue of the anisotropy parameter κ . At small κ < 1/2,the trajectories with infinite growing γ are present, andthe standard separatrix trajectories connect together differentsaddle points, see Fig. 3(c). As mentioned above; only suchtrajectories are present at the limit κ → 0, but, in fact, even for

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FIG. 3. (Color online) Phase plane representations for dissipation-free nonlinear longitudinal spin oscillations for different values of theparameter κ; κ = 0.6, 0.5, and 0.2 for (a), (b), and (c), respectively. The green and red ellipses present the minima and maxima, respectively; thestandard saddle points are depicted by red rectangles, while the standard separatrix trajectories are drawn by red lines. The singular trajectorieswith s = ±1 and the separatrix trajectories entering the nonstandard saddle points on these lines s = ±1 are shown by blue lines in (a) and(c). For the critical value κ = 0.5, all the separatrix trajectories and the singular trajectories with s = ±1 organize a common net; and on thecorresponding frame (b) all of them are presented by red lines.

the small value of κ = 0.2 used in Fig. 2(c), the main part of theplane is occupied by the trajectories that change the spin. Atthe critical value κ = 1/2, the separatrix trajectories connectthe saddle points at s = 0, γ = 0 and the degenerated saddlepoints at the singular lines with s = ±1. At larger anisotropyκ > 1/2, as in Fig. 3(a), the only separatrix loops entering thesame saddle point are present.

The minima, saddle point, and the separatrix are the keyingredients for the switching between the ground states withs = s and s = −s. The case of large anisotropy κ > 1/2 lookslike the standard one for the switching phenomena, whereasfor small anisotropy the situation is more complicated.

C. Damped longitudinal motion

Damping is a crucial ingredient for the dynamical switch-ing between different, but equivalent in energy, states. Thehigh-frequency mode of longitudinal oscillations have high-enough relative damping; as was found from microscopiccalculations,34 the decrement of the longitudinal mode =λωl , where λ ∼ 0.2. To account for the damping in thedynamic equations for s and γ , it is useful to consider adifferent parametrization of the longitudinal dynamics. Letus now introduce a unit vector σ = σ1e1 + σ2e2 + σ3e3 withcomponents

σ3 = s, σ1 = 〈S2y − S2

x〉, σ2 = 〈S1S2 + S2S1〉. (12)

Being written through σ , the equation of motion takes the formof the familiar Landau-Lifshitz equation

h∂σ

∂t= [σ × heff] + R, heff = −∂W

∂σ, (13)

where heff can be treated as an effective field for longitudinaldynamics, and the relaxation term R is added. The equation ofmotion with R = 0 is fully equivalent to the Hamilton form ofthe equation found from (9), but the form of the dissipation is

more straightforward in unit-vector presentation. The choiceof the damping term in a standard equation for the motionof the transverse spin is still under debate.15,36 But here thedamping term can be written in the simplest form, as in theoriginal paper of Landau and Lifshitz, R = λ[heff − σ (heffσ )].The arguments are as follows: (i) this form gives the correctvalue of the decrement of linear oscillations = λωl ; and (ii)it is convenient for analysis, because it keeps the conditionσ 2 = 1. Finally, the equations of motion with the dissipationterm of the aforementioned form are

hds

dt= −∂W

∂γ− λ(1 − s2)

∂W

∂s,

(14)

hdγ

dt= ∂W

∂s− λ

(1 − s2)

∂W

∂γ.

These equations describe the damped counterpart of thenonlinear longitudinal oscillations discussed in the previoussubsection and present as phase portraits in Fig. 3. Thecharacter of the motion at not-too-large λ can be qualitativelyunderstood from energy arguments. The trajectories of dampedoscillations in any point of the phase plane approximately fol-low the nondamped [described by equation W (s,γ ) = const]ones, but cross them passing from larger to smaller values ofW , see Figs. 4 and 6. It happens that for the case of interest, thedynamics is caused by the time-dependent stimulus. An actionof the stimulus on the system can be described by adding thecorresponding time-dependent interaction energy �W to thesystem Hamiltonian W → W (s,γ ) + �W (s,γ,t). Within thisdynamical picture, �W produces an “external force” drivingthe system far from equilibrium.

The analysis is essentially simplified for a pulselike stimu-lus of a short duration �t (much shorter than the period of mo-tion ωl�t � 1). In this case, the role of the pulse is reduced tothe creation of some nonequilibrium state, which then evolvesas some damped nonlinear oscillations described by the “free”

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FIG. 4. (Color online) Phase plane representation of dampedlongitudinal spin oscillations for κ = 0.6. Here and in Fig. 6,the dashed lines (obtained analytically before) represent the phasetrajectories without dissipation, while the full lines are trajectoriesfor dissipation constant λ = 0.2, found numerically. The separatrixlines are drawn by red curves.

Eqs. (14) with �W = 0. The phase plane method, which showsthe behavior of the system for arbitrary initial conditions, isthe best tool for the description of such an evolution.

First, let us start with the analysis for high-enoughanisotropy. The corresponding phase portrait is present inFig. 4. The general property of the phase plane is that thephase trajectories cannot cross each other; they can onlymerge at the saddle points. Thus the trajectories coming todifferent minima are stretched between two separatrix linesentering the same saddle point from different directions, asshown in Fig. 4. From this it follows that any initial state witharbitrary nonequilibrium values of spin s(+0), but withoutdeviation of γ from its equilibrium value, evolve to the statewith the same sign of the spin as for s(+0), and no switchingoccurs. However, if the initial condition is above the separatrixtrajectory, entering the saddle point, the evolution will movethe system to the equivalent minimum with the sign of the spinopposite to the initial one s(+0), realizing the switching.

The switching can be realized for different initial condi-tions, which correspond to a high enough deviation from theinitial state. The smaller deviation of s is present, the largerdeviation of γ is necessary. For the case of high anisotropyκ > κC = 1/2, even if the initial value of s equals to itsequilibrium value s(0) = s = √

1 − κ2, the switching can beobserved, but for quite large values of γ (0). However, the“minimal” deviation of the ground state corresponds to boths(0) < s and γ (0) �= 0. Note that, except some short initialstage, t < (2–4)tex, the behavior of damped oscillations is verycommon for these two regimes. The time evolution for twocharacteristic initial conditions are shown in Fig. 5.

FIG. 5. (Color online) Time evolution of the spin s and quadrupo-lar variables σ1,2, see Eq. (6), describing the switching for the magnetwith κ = 0.6. The data for two characteristic initial conditions,corresponding, to relatively small (or large) initial deviation of s

and relatively large and (or small) initial deviation of γ are presentby full and dash-dot lines, respectively. For concrete calculations,the values s(0) = 0.5, γ (0) = 0.1π and s(0) = 0.1, γ (0) = 0.03π

are used in these cases. The dash lines demonstrate the switchingfor the equilibrium initial value of spin s(0) = √

1 − κ2 = 0.8 andlarge enough γ (0) = 0.19π . The values of spin s (red lines) andσ1 = 〈S2

y − S2x 〉 (blue lines) are going to their finite equilibrium values

−√1 − κ2 and κ = 0.6, whereas σ2 → 0 (green lines).

Figure 6 shows the phase plane for Eqs. (14) for the morecomplicated case of low anisotropy κ < κC = 0.5, demon-strating several possible scenarios of the switching of thesign of the spin value during such dynamics. Note once morethe qualitative difference from the case of small anisotropy:When the anisotropy constant is lower than the critical valueκC = 0.5, the initial deviation of s is necessary for switching,which can be seen comparing Figs. 4 and 6. Here the separatrixtrajectories for the damped motion can be monitored from theirmaxima, and the full picture of the behavior can be understoodonly when including a few equivalent foci with γ = 0, ±π, ± 2π , etc., with different, but equivalent in energy, valuesof the spin s = ±s, s = √

1 − κ2, and different saddle points,located at γ = 0, ± π, ± 2π . As for high anisotropy, thetrajectories evolving to different minima are located betweentwo branches of the separatrix lines, but now this “separatrixcorridor” is organized by separatrix lines entering differentsaddle points. The switching phenomenon is also possible, butthe process involves a few full turns of the variable γ .

The general rule for any anisotropy can be formulated asfollows: For realizing spin switching, one needs to start fromthe states just above the separatrix line entering the saddlepoint from positive values of s. The larger is the deviation ofthe initial value of the spin from the equilibrium, the smallervalue of γ (0) would realize the switching. The asymptoticbehavior of the separatrix trajectories at γ,s → 0, is importantfor this analysis, and can be easily found analytically as

(γ

s

)separ

= Rsepar

= 1

8κ[λ(1 + 3κ) +

√λ2(1 + 3κ)2 + 16κ(1 − κ)].

(15)

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LONGITUDINAL MAGNETIZATION REVERSAL IN . . . PHYSICAL REVIEW B 88, 144420 (2013)

FIG. 6. (Color online) Phase plane for the damped spin evolutions for low anisotropy κ = 0.2. The central frame shows the full diagram;left and right panels demonstrate the details of the behavior near the equilibrium values s = −s and s = s, respectively. On this frame, theregions colored by green and yellow correspond to different basins of attraction with initial values leading to the equilibrium states with s = s

and s = −s, respectively.

From the asymptotic equation (15), the correspondingratio Rsepar = γ (0)/m(0) is smaller for small values of λ;but even when λ → 0, it exceeds the value Rsepar(λ = 0) =0.5

√(1 − κ)/κ . However, for finite damping and small κ → 0,

the value of Rsepar = λ/4κ → ∞, when κ → 0. The separatrixbecomes almost vertical in this limit, and the reversal isbasically prohibited. Thus, the switching could occur fornonzero values of κ > λ ∼ 0.2.

IV. INTERACTION OF THE LIGHT PULSE ON THESPIN SYSTEM: CREATION OF THE INITIAL STATE

FOR SWITCHING

Let us now consider the longitudinal spin evolution causedby a specific stimulus: a femtosecond laser pulse. As men-

FIG. 7. (Color online) The same as in Fig. 5, but for smallanisotropy κ = 0.2. Here the full and dash-dot lines show the behaviorfor minimal deviation s(0) = 0.55, with the large value of γ (0) =0.4π and the small values s(0) = 0.2, γ (0) = 0.1π , respectively. Notehere much larger amplitude of the sign-alternating oscillations of σ1

within the initial stage of evolution, that corresponds with the changeof γ on π , see Fig. 6.

tioned above, the optimal initial state for the switching shouldinclude the deviation of the spin value, “spin quenching,”and, simultaneously, a nonzero deviation of the quadrupolarvariable γ . We now briefly discuss how both conditions couldbe realized.

A. Spin quenching

In the literature, the laser-induced ultrafast quenching of themagnetic moment is mainly associated with transition metalsor their alloys.13,37–39 Moreover, the microscopic scenario ofthis effect is based on d-shell itinerant electrons.40 (Note herethe recent work41 where a scenario for such a quenching hasbeen proposed for rare-earth metals.) However, the theorydeveloped here can be applied to different materials, eitherconducting or insulating. The quenching of the magneticorder is known for many nonmetallic compounds, insulators,and semiconductors. Note the observation of this effect fora number of spinel-type chromium chalcogenides,42 eithermetallic or insulating, e.g., insulating ferrimagnetic FeCr2S4,CoCr2S4, and insulating ferromagnetic CdCr2S4. It has beenmentioned that the quenching becomes faster for magnets withhigh anisotropy, which are interesting for our consideration.Femtosecond demagnetization has also been found for fer-romagnetic semiconductors InMnAs and GaMnAs.43,44 Thelarge magneto-optic response, witnessing a drastic change ofthe order parameter within the first few hundreds of femtosec-onds, has been observed for insulating antiferromagnets nickeloxide nickel NiO (Ref. 8) and Cr2O3.45 A fast (over a timescale of 300 fs) and strong suppression of the magnetic orderunder the action of a femtosecond laser pulse has been foundrecently in copper oxide CuO.46 Whereas the mechanismsof these effects is not as clear as for transition metals, theirexistence for different compounds is undoubtful. Note aswell the recent observation of femtosecond suppression ofthe antiferromagnetic order in manganite Pr0.7Ca0.3MnO3,exhibiting colossal magnetoresistance.47

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B. Control of quadrupolar variables

The next nontrivial question is: How can we create adeviation of the quadrupolar variable γ from its equilibriumvalue? To find out how to do this, let us consider possiblemechanisms of light interaction with quadrupolar variables ofthe magnets with nonsmall single-ion anisotropy.

The interaction of the spin system of magnetically orderedmedia and light is described by the Hamiltonian (as above,written per spin) �W = εij v0Ei(t)E∗

j (t)/16π , where v0 is thevolume per spin, Ei(t) is the time-dependent amplitude ofthe light in the pulse, εij = d(ωε

(spin)ij )/dω, ε

(spin)ij is the spin-

dependent part of the dielectric permittivity tensor, and ω is thefrequency of light. For the longitudinal dynamics consideredhere, circularly polarized light propagating along the z axisacts on the z component of the spin via the standard inverseFaraday effect, with the antisymmetric part of ε

(a)ij as ε(a)

xy =−ε(a)

xy = sαF, giving an interaction of the form

�Wcircular = sαFv0

16π|Ecirc|2σ, (16)

where Ecirc is the (complex) amplitude, and σ describesthe pulse helicity: σ = ±1 for right-handed and left-handedcircularly polarized laser pulses. To describe qualitatively theresult of the action of the light pulse, let us now assume thatthe pulse duration τpulse is the shortest time of the problem. Ifthe pulse duration is shorter than the period of spin oscillations,the real pulse shape can be replaced by the Dirac δ function|Ecirc|2 → E2

pτpulseδ(t), where E2p = ∫ |Ecirc|2dt/τpulse char-

acterizes the pulse intensity. (Note that this approximationis still qualitatively valid even for any comparable values ofτpulse and 2π/ω.) Then, using Eqs. (14) one can find the effectproduced by the pulse. Within this approximation, the actionof a pulse leads to an instantaneous deviation of the variable γ

from its equilibrium value, which then evolves following thenonperturbed equations of motion (14). Keeping in mind thatbefore the pulse action the system is in equilibrium, s(−0) = s

and γ (−0) = 0, it is straightforward to find the values of thesevariables (s and γ ) after the action of the pulse s(+0) andγ (+0). For our purposes, the nonequilibrium value of thequadrupolar variable γ is important:

γ (+0) = − αF

16πhE2

pv0τpulseσ. (17)

The cumulative action of the circularly polarized pulse,including an essential reduction of the spin value (caused eitherby thermal or nonthermal mechanisms) and the deviation of γ

described by (17) could lead to the evolution we are interestedhere, switching the spin of the system. Note that the effectivefield associated with the inverse Faraday effect for transparentinsulating magnets reaches a few Tesla.2 The theory of theultrafast inverse Faraday effect for metals has been developedby Popova and coauthors48 and the experiment shows hugevalues, up to 10 T for GdFeCo, see Fig. 2 of Ref. 49.Note here that, for standard spin reduction, the polarizationof the light pulse is not essential,37–39 whereas the valuesof γ (+0) are opposite for right- and left-handed circularlypolarized pulses. These features are characteristic of the effectdescribed here. Note the recent experiment where the role ofcircular polarization in spin switching for GdFeCo alloy was

mentioned, but the authors49 have attributed it to magneticcircular dichroism.

V. CONCLUDING REMARKS

First, let us now compare the approach developed in thisarticle with previous results on subpicosecond spin evolution.The first experimental observation of demagnetization forferromagnetic metals under femtosecond laser pulses showsthat the magnetic moment can be quenched very fast tosmall values, much faster than 1 ps.37–39 These effects areassociated with a new domain of the physics of magnets,femtomagnetism,50 and its analysis is based on the micro-scopic consideration of spins of atomic electrons,51,52 oritinerant electrons.40 Not discussing this fairly promising andfruitful domain of magnetism, note that, to the best of ourknowledge, no effects of magnetization reversal during this“femtomagnetic stage” has been reported in the literature (see,however, the recent article47). For example, the subpicosecondquenching processes for the ferromagnetic alloy GdFeCo areresponsible for the creation of a far-from-equilibrium state,but the evolution of this state, giving the spin reversal, can bedescribed within the standard set of equations for the sublatticemagnetizations.16

In contrast, here we propose some pathway to switch thesign of the magnetic moment during extremely short times, ofthe order of the exchange time. It is shown here that the spindynamics for magnets with nonsmall single-ion anisotropy canlead to the switching of the sign of the magnetic moment viathe longitudinal evolution of the spin modulus together withquadrupolar variables (concretely, the quantum expectationvalues of the operators bilinear over the spin components Sx

and Sy). It is worth to stress here that the “restoring force” forthis dynamics is the exchange interaction, and the characteris-tic time is the exchange time. However, to realize this scenario,one needs to have a significant coupling of the spin dipolevariable s and spin quadrupole variables. Obviously this effectis going beyond the standard picture of spin dynamics based onany closed set of equations for the spin dipolar variables alone.Note that our approach based on the full set of variables for theatomic spin is “more macroscopic” than the “femtomagnetic”approach,40,51,52 dealing with electronic states.

We have found this type of switching within the concretemodel (1) with spin one, Heisenberg exchange, and strongenough single-ion anisotropy. Such an anisotropy is knownfor the numerous magnets based on anisotropic ions oftransition elements such as Ni2+, Cr2+, Fe2+. Note the firstspin-one system CsNiF3 investigated thoroughly with respectto its nonlinear properties,53 with the ratio of the anisotropy(estimated with quantum corrections) to exchange integralas high as 0.3. A better candidate is nickel fluosilicate hex-ahydrate NiSiF6·6H2O, with spin-one Ni2+ ions, coupled bythe isotropic ferromagnetic exchange interaction and subjectto high single-ion anisotropy. For this compound, the strongeffect of the quantum spin reduction is known, with its strengthdependent on the pressure: The value of K/J is growing withthe pressure P resulting in the value 〈S〉 = 0.6 at P = 6 kbarand leading to the transition to the nonmagnetic state with〈S〉 = 0 at P ∼ 10 kbar.54,55 Other necessary conditions are thefollowing: The possible materials should be very susceptibleto magnetization quenching, as well as they should have a

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sizable Faraday effect. Unfortunately, we are not aware of thecorresponding properties for the aforementioned materials.

A number of recent experiments were done with rare-earthtransition-metals alloys.13,14,49,56 Note here a rich varietyof nonlinear spin dynamics observed for thin films of theFeTb alloy under the action of femtosecond laser pulses.56

However, the theory developed here for a simple one-sublatticeferromagnet cannot be directly applied for the description ofsuch compounds. Ferromagnetic order with high easy-planeanisotropy is present for many heavy rare-earth elements, suchas Tb and Dy at low temperatures.57 This feature is known bothfor bulk monocrystals,57 and for thin layers and superlattices,see Ref. 58 and references wherein. Rare-earth metals havenonzero values of both spin and orbital momentum, formingthe total angular momentum of the ion, and for their descriptionthe theory needs some modifications.

We conclude this article with some discussions of possiblegeneralizations of the model and also pointing out some othermagnets which probably could be used for realizing suchkind of dynamics. Note first that our model neglects somephysical interactions, for example, magnetoelastic coupling,which can be strong for magnets with high anisotropy. Thedetailed discussion of such effects is far beyond the scopeof our article. Note only that many of the previous theories(either numerical or analytical) describing the laser-inducedspin reorientation for GdFeCo alloys also do not considerthis interaction,13,14,16,41 whereas the rare-earth compoundsare associated with giant magnetostriction.57 In this workonly spin-one ferromagnets with high single-ion anisotropywere considered. Nevertheless, we believe that the switchingpathway presented here is more general than the concretemodel discussed in this work

Going beyond the concrete model (1), let us stress thatthe main necessary conditions for the realization of thistype of dynamics is the presence of both nonsaturated valueof spin (quantum spin reduction) and significant values ofnondipolar (quadrupolar) variables in the ground state, whichare coupled. As a consequence of their coupling, the longi-tudinal dynamical mode with an exchange “restoring force”appears. All of the above features are characteristics of theso-called ferromagnetic-quadrupolar states, known for manyrare-earth and actinide compounds with nonconventionalmagnetic order, see the review articles.25,26 The cause of suchstates and modes cannot only be single-ion anisotropy, but non-Heisenberg exchange interaction as well. Such interactions

include biquadratic exchange (�S1 · �S2)2, and for higher spinsS > 1 the terms (�S1 · �S2)n, with 2 < n � 2S can be presentas well. All these interactions could lead to spin reductionand the formation of ferromagnetic-quadrupolar states. Theyalso couple the spin dipole and multipole variables, givingrise to longitudinal modes. The above features are knownfor either integer spin-two models,59 or half-odd three-halvesspin models.20,33 Moreover, the Hamilton structure of thedynamical equations for these coupled variables is expectedto be the same as for spin-one systems (10) [of course, theform of the Hamiltonian can differ from the one in Eq. (9)].The reason is the following: From both classical and quantummechanics it is known that the angular momentum and theangle of rotation around its direction are Hamilton-conjugatedvariables. Thus one can expect the possibility of the modewith coupled rotation of the quadrupolar ellipsoid around itsprincipal axis, coinciding with the spin direction, and theoscillation of the spin value along this direction. Of course,this is only a qualitative argument. A detailed analysis ofconcrete models with spin S > 1 is of interest, but it is farbeyond the scope of this article. The effects of spin switchingcaused by dipolar-quadrupolar longitudinal spin dynamics canbe present for magnets with high values of the atomic angularmomentum, which are in ferromagnetic-quadrupolar states. Itis difficult to identify the best candidate for the realizationof these effects of spin switching among the large variety ofnonconventional magnets. A possible candidate could be theintriguing URu2Si2, which has been a puzzle for decades.26

This material has a phase transition from a magnetic phaseto a phase with so-called hidden order,60 and a mode withlongitudinal spin oscillations in the vicinity of this transition.61

ACKNOWLEDGMENTS

This work is partly supported by the Presidium of theNational Academy of Sciences of Ukraine via Projects No.VTs/157 (E.G.G.) and No. 0113U001823 (B.A.I.) and bygrants from the State Foundation of Fundamental Researchof Ukraine No. F33.2/002 (V.I.B. and Yu.A.F.) and No.F53.2/045 (B.A.I.). F.N. acknowledges partial support fromthe ARO, RIKEN iTHES Project, MURI Center for DynamicMagneto-Optics, JSPS-RFBR Contract No. 12-02-92100,Grant-in-Aid for Scientific Research (S), MEXT Kakenhi onQuantum Cybernetics, and Funding Program for InnovativeR&D on S&T (FIRST).

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